People have claimed many things about Godel's theorem and so you will find many mathematicians reacting to them by taking a minimalist view of how the theorem can be applied. I understand their reaction against many of the claims, but I don't agree with the minimalist view because I am a theoretical physicist and have been infected with the desire to describe nature as a mathematical system. I am a realist and a believer that mathematics and its arithmentic are not merely constructs of the human mind but integral to reality in that we apprehend them in physics. Physics as physicists conceive of it is a Turing machine in the sense that nature performs sequential computations through time in each locality of space (or nonlocally in view of QM) according to well-defined "laws", which are reducible to a minimum set. This set of laws (and I am talking about the real laws that govern nature and not just our attempted description of them) acts like "axioms" that define how natur
e computes the state of the universe at the next moment of time as a deduction from the present moment of time. (I would like to modify that statement to take out the arrow of time, but that would muddle things.)

You wrote about Godel's theorem:

> In fact, Godel's theorem
> cannot be solved by just adding more and more complexity, even in
> infinity, the problem still arises.

That is correct if we are only discussing countable infinities (cardinality aleph-null), and that is why I used the adjective "countable". It is why the description of Godel's theorem that you cited (not repeated here) uses the term "computationally enumerable," which is a synonym for "countable". It is true that in a countable infinity, the problem does indeed still arise -- a countably infinite set of axioms will still be incomplete. But Godel's theorem does not prove that there is any problem with completeness in a system with an uncountable infinity of axioms (a higher cardinality), or in a system not reducible to axioms. A realist view of physics, if it sees arithmetic as being a part of physics and hence a part of reality, must conclude that nature cannot be reduced to a countable set of axioms, not even a countably infinite set. Thus, (in that view of reality) reductionist science must fail at some point, and Dawkins' belief that everything arises from something s
impler cannot be true; whatever lies behind reality must in the final analysis be infinite to a cardinality greater than aleph-null or must be irreducible to simple laws. This does not say that anything within nature taken individually cannot be reducible to a finite number of laws and Turing-type computations in time (e.g., the formation of a star is described simply by physical laws), or that the ID arguments are necessarily wrong. It only says that whatever lies behind reality and gives the arithmetic to nature is irreducible. Also, this does not prove that whatever lies behind reality is a Mind, but let me say more...

You wrote:

> Of course completeness can be obtained at the cost of
> consistency, or consistency comes at the cost of completeness, or
> perhaps the mind is not a Turing machine.

This is not really correct. Completeness and consistency are not two alternatives that allow us to successfully obtain one of them. They are both unreachable according to Godel's theorem for an axiomatic system containing arithmetic and defined by a countable set of axioms. (Your quoted source was rather muddled on this point.) Completeness cannot be obtained at the cost of consistency. (In an inconsistent system "completeness" becomes even more unobtainable in the sense that it becomes meaningless.) The answer to completeness (but not to consistency) is that a complete system, if such can exist, must not be reducible to axioms; or if it does derive from axioms, then it must have an uncountably infinite number of axioms (cardinality greater than aleph-null).

Similarly, the way to obtain consistency is not to choose it as an alternative to completeness, because according to Godel's theorem even an incomplete system cannot be proven to be consistent by any countable set of deductions. (Note: a finite number of Turing machines operating for eternity can produce only a countable set of computations.) Thus, if a consistent system does exist, then it must not be reducible to axioms, or else if it does derive from axioms then it must have been used to complete an uncountably infinite number of computations to discover whether its axioms are consistent. (In an axiomatic system, the method of doing the computations is defined by the axioms.) So if physics is real and reducible to axioms, contains arithmetic, and cannot exist in an inconsistent state, then something must have done an uncountably infinite number of computations before the physics can exist. We can also apply this to God or anything else further back behind physics, in
the style for which you praise Dawkins. But how can anything's existence depend upon an outcome of its existence? I see in this a statement that nature (or God, or whatever lies furthest back without an infinite regress) is not reducible and was not simple at its origin. Either physics is not reducible, or God is not reducible, etc.

This is not a complete argument that whatever lies behind reality must be a Mind, but doesn't this begin to strike you that believing in an irreducible Being like God as opposed to reductionist materialism is not foolishness?

I believe the kind of self-existent "complexity" in God's nature implied by this argument still fits the theologian's definition of divine "simplicity". In its essence the entity behind nature still has no parts and is indivisible. But it is complex in that it must exist with knowledge of the outcome of an uncoutably infinite number of computations as an essential characteristic. This kind of "complexity" theologians do admit within divine simplicity. So I think Dawkins and Platinga may be talking past each other on this point of complexity versus simplicity. God is simple in essence but his knowledge contains everything, and so in Dawkins' language that would mean He is complex.

We can probably hypothesize a basis for reality that is not a Mind, but which has computed an uncountably infinite number of deductions to find the consistent (and uncountable) set of axioms that allows reality to be both consistent (and complete). Perhaps some analog to QM can perform this miracle by finding zero probability in the uncountably infinite set of inconsistent descriptions for itself -- the miracle of quantum computing! But it is strange that such a system must be defining **itself**, not something else, while doing those computations, and so it must perform the uncountable number of calculations before it can exist to perform the calculations. Or maybe the parts of this Thing that do the calculations with inconsistent laws fail to materialize precisely because they are inconsistent. How can we ever penetrate this? You are right that this does not prove that an infinite Mind exists, such as we may call "God". However, I did include the three letters "IMO" b
ecause I feel that it is more reasonable to call such a basis for reality "a Mind" rather than "physics" and to believe that it is probably a Being rather than a thing. (And there is more to this argument than I have stated here.)

> As far as the concept of eternal, I believe the Greeks had Gods which
> were not eternal, some were born, others died.

Precisely, and that is what we find it in almost all polytheistic religions. The gods emerged from chaos in a way that could (with some stretching) be made out as consistent with Dawkins' world view. But Dawkins is very clear that he is not addressing these polytheistic religions, but only the monotheistic faiths that look back to Abraham: Judaism, Christianity, and Islam. All of these hold that God is self-existent, "I AM that I AM. Tell them that I AM has sent you."

God bless,
Phil

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Received on Wed May 9 01:48:29 2007